Find the equation of the straight line passing through the origin and through the point of intersection of the lines 5x + 7y = 3 and 2x - 3y = 7.

5x + 7y = 3 ….(i)

2x - 3y = 7 ….(ii)

Multiply (i) by 3 and (ii) by 7,

15x + 21y = 9 ….(iii)

14x - 21y = 49 ….(iv)

Adding (iii) and (iv) we get,

⇒ 29x = 58
⇒ x = 2.

Substituting x = 2 in (i), we get

⇒ 5(2) + 7y = 3
⇒ 10 + 7y = 3
⇒ 7y = 3 - 10
⇒ 7y = -7
⇒ y = -1.

Hence, the point of intersection of lines is (2, -1).

The equation of the line joining (2, -1) and (0, 0) will be given by two-point form i.e.,

$y - y$1 = \dfrac{y2 - y1}{x2 - x1}(x - x1)

Putting values in above equation we get,

$\Rightarrow y - (-1) = \dfrac{0 - (-1)}{0 - 2}(x - 2) \\[1em] \Rightarrow y + 1 = \dfrac{1}{-2}(x - 2) \\[1em] \Rightarrow -2(y + 1) = x - 2 \\[1em] \Rightarrow -2y - 2 = x - 2 \\[1em] \Rightarrow x + 2y - 2 + 2 = 0 \\[1em] \Rightarrow x + 2y = 0.$

Hence, the equation of the line is x + 2y = 0.